Melt transport from partial melt sources in the asthenosphere through the lithosphere and crust involves many complexities. Reactive and rheological instabilities lead to strongly localised flow of melt through a deforming and reacting host rock. The resulting regimes of magmatic activity involve a wide range of time and length scales. Numerical modelling offers promising avenues of quantitative research into these phenomena. Yet, the complex nature of melt transport processes poses challenging computational problems requiring robust and efficient numerical techniques. Here, I will present some recent developments in computational magma dynamics, in terms of physical model description, computational methods, and scientific applications. The latter include melt transport across the flow-to-fracture transition for intraplate magmatism in a continental lithosphere, as well as volatile flux-driven reactive melt transport beneath mid-ocean ridges. These examples illustrate both opportunities and challenges for computational modelling of melt transport from source to surface.

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